Indian Institute of Technology Bombay GIS-based mineral resource potential mapping - Modelling...

Indian Institute of Technology BombayIndian Institute of Technology Bombay

GIS-based mineral resource potential mapping - Modelling approaches

Exploration datasets with homogenous coverage – required for all models

Expert knowledge (a knowledge base) and/or

Model parameters estimated from mineral deposits data(Known deposits required)Brownfields explorationExamples - Weights of evidence, Bayesian classifiers, NN, Logistic Regression

Data-driven Knowledge-driven

Hybrid

Mineral deposit data

Model parameters estimated from both mineral deposits dataand expert knowledge (Known deposits necessary)Semi-brownfields to brownfields explorationExamples – Neuro-fuzzy systems

Training data Expert knowledge

Model parameters estimated from expert knowledge (Known deposits not necessary)Greenfields explorationExamples – Fuzzy systems; Dempster-Shafer belief theory


2

Structure of a model for mineral resource potential mapping

∫Integrating function

• linear or non-linear

• parameters

Input predictor maps

• Categoric or numeric

• Binary or multi-class

Output mineral potential map

• Grey-scale or binary


Probabilistic Model (Weights of Evidence):

- used in the areas where there are already some known deposits

- spatial associations of known deposits/oil well with the geological features are used to determine the probability of occurrence of a mineral deposit (or well) in each unit cell of the study area.

Fuzzy Model:

- used in the areas where there are no known mineral deposits

- each geological feature is assigned a weight based on the expert knowledge, these weights are subsequently combined to determine the probability of occurrence of mineral deposit in each unit cell of the study area.

GIS MODELS FOR MINERAL EXPLORATION

Indian Institute of Technology Bombay

Fuzzy Inference Systems for Mineral Prospectivity Mapping


Artificia

l inte

lligence

!


IntroductionFuzzy logic:

• A way to represent imprecision in logic and approximate reasoning• A way to make use of natural language in logic

•Humans say things like :• “If it is cool and dry, I will walk faster on my morning walk”•"If it is overcast and warm and humid, it will rain heavily"

•Linguistic variables:• Speed: {Fast, slow}•Temp: {freezing, cool, warm, hot}•Cloud Cover: {overcast, partly cloudy, sunny}•Humid: {high, average, low}•Rain: {Heavy, moderate, light}


Crisp (Traditional) Variables

• Crisp variables represent precise quantities:Temp = 36 deg CHumidity = 70%

• Some thing like:– If the cloud cover is 90% and temperature is 40

degrees C and humidity is 70%, then the rainfall would be 30 mm.

– If the temperature is 25 deg C, I will walk at 20 km/hr on my morning walk.


Fuzzy Sets• Extension of Classical Sets• Not just a membership value of in the set and out the set, 1 and 0

– but partial membership value, between 1 and 0


Example: Height

• Tall people: say taller than or equal to 6 feet– 6’, 6’1”, 6’3” feet are members of this set– 5’11.9” are not members of this set - Is that

reasonable? (measurement can be inaccurate and/or imprecise )


Example: Weekend days


In fuzzy logic, the truth of any statement becomes a matter of degree.

Q: Is Saturday a weekend day?

A: 1 (yes, or true)

Q: Is Tuesday a weekend day?

A: 0 (no, or false)

Q: Is Friday a weekend day?

A: 0.8 (for the most part yes, but not completely)

Q: Is Sunday a weekend day?

A: 0.95 (yes, but not quite as much as Saturday).

BINARY FUZZY


Example: Season

BINARY FUZZY

Summer Rainy Autumn Winter Summer Rainy Autumn Winter


MEMBERSHIP FUNCTION

A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1.

Example: Membership function of a set of tall people

Crisp set

Fuzzy set:

6x1μ

6x0μ

x

x

6 if 1

56 if 5

5 if 0

x

xx

x

x

x

x


Example: Height


Membership Function

A = {x | x > 6}

A = {x, µA

(x) | x ∈ X}

A membership function must vary between 0 and 1.

The function itself can be an arbitrary curve whose shape we can define as a function that suits us from the point of view of simplicity, convenience, speed, and efficiency.

A classical set might be expressed as

A fuzzy set is an extension of a classical set. If X is the universe of discourse and its elements are denoted by x, then a fuzzy set A in X is defined as a set of ordered pairs.

µA(x) is called the membership function (or MF) of x in A. The membership function maps each element of X to a membership value between 0 and 1.

µA(x) is called the membership function (or MF) of x in A. The membership function maps each element of X to

a membership value between 0 and 1.


Membership functions• piece-wise linear functions

• the Gaussian distribution function• the sigmoid curve

• quadratic and cubic polynomial curves


Membership Functions

• For each variable value a different membership function is required

• Temp: {Freezing, Cool, Warm, Hot}

50 70 90 1103010

Temp. (F°)

Freezing Cool Warm Hot

0

1

5 20 30 40Temp OC


Fuzzy Operators• How do we use fuzzy membership functions in predicate logic?• Fuzzy logic Connectives:

– Fuzzy Conjunction, – Fuzzy Disjunction,

• Operate on degrees of membership in fuzzy sets


Fuzzy Disjunction (= OR Operator)

• AB max(A, B)• AB = C "Quality C is the disjunction of Quality A and

B"

0

1

0.375

A

0

1

0.75

B

(AB = C) (C = 0.75)


Fuzzy Conjunction (=AND Operator)

• AB min(A, B)• AB = C "Quality C is the conjunction of Quality A and

B"

0

1

0.375

A

0

1

0.75

B

(AB = C) (C = 0.375)


Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0

1

A

0

1

B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40


Example: Fuzzy Conjunction

Calculate AB given that A is .4 and B is 20

0

1

A

0

1

B

.1 .2 .3 .4 .5 .6 .7 .8 .9 1 5 10 15 20 25 30 35 40

Determine degrees of membership.


Subjective assignment of fuzzy membership values

• It is also possible to assign the fuzzy membership values subjectively (without using a membership function)

Lithotype

Membership value

Granite 0.2

Dolerite 0.7

Magnetite quartzite

0.9

Diorite 0.4

Distance to a fault

Membership value

0 – 1 km 0.9

1 – 2 km 0.7

2 – 3 km 0.5

3 – 4 km 0.3

4 – 5 km 0.1

> 5 km 0.001


Fuzzy if-then rulesFuzzy if-then rule statements are used to formulate the conditional statements that comprise fuzzy logic.

A single fuzzy if-then rule assumes the form:

if x is A then y is B

where A and B are linguistic values defined by fuzzy sets on the ranges X and Y, respectively.

The if-part of the rule "x is A" is called the antecedent or premise

The then-part of the rule "y is B" is called the consequent or conclusion.


if x is A then y is B

For example: If FeO is high then gold potential is average

“High” is represented as a number between 0 and 1, and so the antecedent is an interpretation that returns a single number between 0 and 1.

“Average” is represented as a fuzzy set, and so the consequent is an assignment that assigns the entire fuzzy set B to the output variable y.

Fuzzy if-then rules


The input to an if-then rule is the current value for the input variable (in this case, FeO) and the output is an entire fuzzy set (in this case, gold potential).

The consequent specifies a fuzzy set be assigned to the output.

The implication function then modifies that fuzzy set to the degree specified by the antecedent. The most common ways to modify the output fuzzy set are truncation using the min function, where the fuzzy set is truncated.

This set has to be defuzzified, assigning one value to the output.

40% 50% 60% 70%

0

1

1 tonne 1000 tonnes

0

1

0.3

Fuzzy if-then rules

FeO Gold


Fuzzy if then rules can be quite complex:

IF sky is gray AND wind is strong AND humidity is high AND temperature low, THEN rainfall will be heavy.

IF Granite is Proximal AND Fault is Proximal AND FeO is high AND SiO2 is low, THEN Gold potential is high.

Several fuzzy if-then rules are combined to generate a Fuzzy Inference System

Fuzzy if-then rules & inference systems


Fuzzy Inference SystemStep 1: Identify the factors that control a system:

For example:

Formation of a deposit (or mineral potential of an area) depends on the following factors:

1. FeO content of the rocks

2. SiO2 content of the rocks

3. Closeness (or proximity) to granite

4. Closeness (or proximity) to faults

Step 2: Identify the variables for each of the factor:

1. FeO content: {high, average, low}

2. SiO2content: {high, average, low}

3. Proximity to granite :{proximal, intermediate, distal}

4. Proximity to faults: {proximal, intermediate, distal}

Step 3: Identify the output variable of the system

Mineral potential: {high, average, low}


Fuzzy Inference System Step 4: Decide a fuzzy membership function for each variable

30% 50% 70%0

1High

30% 50% 70%0

1Average

FeO

40% 55% 70%0

1High

40% 55% 70%0

1Average

SiO2

30% 50% 70%0

1Low

40% 55% 70%0

1Low



(Contd.)

Granite

0 km 10 km20km

0

1Proximal

0

1Intermediat

e

0 km 10 km20km

0

1Distal

0 km 10 km20km

Fault

0 km 5 km20km

0

1Proximal

0

1Intermediat

e

0 km 5 km20km

0

1Distal

0 km 5 km10km



(Contd.)

Mineral potential

0

1Low

0

1Average

0

1High

0t 100t 1000t

0t 100t 1000t

0t 100t 1000t


Fuzzy Inference System

Step 5: Develop a set of fuzzy if-then rules to explain the behavior of the system

Rule1: IF FeO is high AND SiO2 is low AND Granite is proximal AND Fault is proximal, THEN mineral potential is high.

Rule 2: IF FeO is average AND SiO2 is high AND Granite is intermediate AND Fault is proximal, THEN mineral potential is average.

Rule 3: IF FeO is low AND SiO2 is high AND Granite is distal AND Fault is distal, THEN mineral potential is low.


Input data preparationStep 6: Rasterize the input predictor maps, combine them, and generate feature vectors

Input feature vector

[3, 8, 33, 800]GIS raster layers

MgO%

Rock type

FeO%

Distance to Fault


Step 7: Represent fuzzy if-then rules in terms of membership functions

1: IF FeO is high & SiO2 is low & Granite is prox & Fault is prox, THEN metal is highImplication (Max)

0

1

0

1

=

2: IF FeO is aver & SiO2 is high & Granite is interm & Fault is prox, THEN metal is aver

30% 50% 70%

0

1

40% 55% 70%

0 km 10 km 20km

0 km 5 km 10km

0t 100t 1000t

3: IF FeO is low & SiO2 is high & Granite is dist & Fault is dist, THEN metal is low

FeO = 60% SiO2 = 60% Granite = 5 km Fault = 1 km Metal = ?

0t 100t 1000t

=

=


Step 8: Combine outputs of each rule

Rule 1: Rule 2: Rule 3: Aggregate (Max)

+ + =

Defuzzify (Find centroid)

125 tonnes metal

Formula for centroid

n

ii

n

iii

x

xx

0

0

)(

)(

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